#483479

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28)(20,25)(21,34)(22,37)(23,27)(24,32)(26,35)(29,41)(30,42)(31,43)(33,48)(36,38)(39,46)(40,45)(44,47);(17,19,29,45,22,33,21,32,23,30,35,47,20,31,36,46)(18,24,38,48,25,40,26,28,27,39,34,43,37,44,41,42);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(19,30)(24,39)(28,42)(31,33)(32,46)(40,44)(43,48)(45,47);(3,9)(6,12)(10,15)(13,16)(17,20,23,22)(18,25,27,37)(19,31,30,33)(21,29,36,35)(24,40,39,44)(26,34,41,38)(28,43,42,48)(32,45,46,47)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,d^c=dh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390675);